0
votes
1answer
32 views

Geometric meaning of $z=\frac{x_1+x_2}{2}$, $y=\frac{x_1+x_2}{|x_1+x_2|}$, and of a substitution (in the complex plane)

Let $S$ be a unitary (that is, $r=1$) circle centered in the origin of the complex plane. Let $x_1 \in S$ and $x_2 \in S$ be the vertices of the triangle $Ox_1x_2$. What is the geometric meaning of ...
2
votes
1answer
31 views

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge to $w_0$ and $n \rightarrow \infty$? I would say ...
0
votes
1answer
18 views

Solve the integral with complex number and floor function.

let $z\in\mathbb{C},\,0<\left|z\right|<1$.I would like to calculate the integral $$\int_{1}^{\infty}\frac{\left(1-z\right)^{\left\lfloor t\right\rfloor }}{t^{2}}dt$$ where ${\left\lfloor ...
2
votes
1answer
33 views

how evaluate $\int_0^{\pi}\frac{1}{(a+\cos{\theta})^2}, a>1 $, using residues theorem?

how evaluate $\int_0^{\pi}\frac{1}{(a+\cos{\theta})^2}, a>1 $, using residues theorem? This problem is an exercise book Complex Analysis of Conway.
3
votes
0answers
33 views

Is the exponential function the one this problem is hinting at?

Suppose that $f$ is holomorphic on all of $\mathbb{C}$ and that $$\lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^nf(z)$$ exists, uniformly on compact sets, and that this limit ...
0
votes
0answers
24 views

Consider$\ f$, defined for all complex numbers except$\ x_0$. What can(not) happen to the real part of$\ f\left(x_0+bi\right)$ as$\ b\to0$?

Yes,$\ f$ is holomorphic. In other words, given that$$\ \lim_{x\to x_0}f(x)=\infty,$$what do we know a priori about $\ \lim_{b\to0}\Re\left(f\left(x_0+bi\right)\right)$? Can it be any extended real ...
0
votes
1answer
31 views

Show that $\frac{1}{2\pi} \int_0^{2\pi} e^{ik\theta}d\theta = 1$ or $0$ depending on $k$.

I'm asked the problem (restating from the question title), $$\textrm{Show that }\frac{1}{2\pi} \int_0^{2\pi} e^{ik\theta}d\theta = \begin{cases} 1,\, k=0\\ 0, \, k\neq0 \end{cases}$$ My Attempt: ...
10
votes
2answers
227 views

Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. ...
0
votes
1answer
55 views

complex differentiation

It is my first text here. So I have started to look at complex numbers in death. I do Uni know, so adding $3+4i$ and $4+7i$ is now nothing. What I am stuck on is the idea of taking a derivative of a ...
0
votes
2answers
44 views

Use mathematical induction to prove Σ n,k=1 (1/k(k+1)) = (n/n+1) for all n in Natural numbers?

This is how far I can get: p(n): nΣk=1 (1/k(k+1)) = (n/n+1) p(1): 1Σk=1 (1/(1+1)) = (1/1+1) => 1/2 = 1/2 p(1) is true. Assume that p(k) is true. p(k) = kΣk=1, (1/k(k+1)) = k/k+1 Show ...
13
votes
1answer
195 views

Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
1
vote
3answers
43 views

Biconditionality-ish of Epsilon Delta Proof of a Limit?

I have known the precise ($\epsilon$, $\delta$) definition of a limit, $$\lim_{x\rightarrow a} f(x) = L \iff \forall \epsilon>0,\exists\delta>0 : (0<|x-a|<\delta \implies ...
0
votes
1answer
24 views

Power series and zeros

When is a power series equal to zero? Example: Take $\sum_{n=0}^\infty a_n(z-z_0)^n$. Is this power series equal to zero only at $z=z_0$ if we assume that we have infinitely many nonzero $a_n$? ...
1
vote
2answers
26 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
0
votes
0answers
38 views

Roots of polynomial

I came across when reading paper: Given $f'(z)+\alpha zf''(z) + \gamma z ^2f'''(z) $ where $\mu = \tfrac{(\alpha-\gamma)-\sqrt{(\alpha-\gamma)^2-4\gamma}}{2}$,$\quad$ $\nu+\mu=\alpha-\gamma$, ...
1
vote
1answer
56 views

Engineer searching for calculus and complex analysis books without limits

I am an engineer and I need to study calculus and complex analysis without too much limits or Riemann sums or proofs. I mean on the differentiation and integration levels and higher (not digging ...
1
vote
0answers
37 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
3
votes
1answer
112 views

On a property of polylogarithm

I have an observation, and I don't know that the following statement is true or not. If not give a counterexample, if it is true prove it, or give a reference about it. Let $n \in \mathbb{R}$, $z \in ...
2
votes
2answers
89 views

Compute $\Gamma(2.7)$

All I know of the Gamma function is $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}e^{-x}\text{ d}x$$ and the recursive formula $$\Gamma(\alpha) = (\alpha-1)\Gamma(\alpha - 1)\text{.}$$ A ...
13
votes
2answers
170 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
7
votes
0answers
106 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
5
votes
4answers
225 views

Wolfram alpha says that $\int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$

Wolfram alpha says that $$ \int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$$ holds. But it has two different values ($\sqrt{i}$). How should I understand this?
0
votes
0answers
50 views

Complex Gaussian Integral - $\int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}}$?

I found some formulas on books, especially the complex gaussian integral formula: $$ \int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}} $$ for $p,c\in\mathbb C$. Then if $p=i=\sqrt{-1}$, the ...
11
votes
5answers
185 views

Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$

Do you know any nice way of expressing $$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$ ? Some simple manipulations involving the integrals lead to an expression that also uses the hypergeometric series. ...
0
votes
0answers
52 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
2
votes
1answer
34 views

About the definition of complex multiplication

Some people say that the complex product is the way it is to respect the distributive law of multiplication. However, the distributive law acts in the whole number, like: $$(a+b)(c+d) = ac + ad + bc ...
0
votes
1answer
84 views

Can a finite value for $\int_1^\infty \exp(x^2)\,dx$ be defined?

Why should $$\int_1^{\infty}\exp(ix^2)dx,\int_1^{\infty}\exp(-ix^2)dx,\int_1^{\infty}\exp(-x^2)dx$$ converges but not: $$\int_1^{\infty}\exp(x^2)dx$$ Is there any way that assigns a value to ...
1
vote
1answer
35 views

Complex roots (review) (advise)

I have to find the complex roots and want a review of my procedure to see if is correct A. $$\sqrt{3i}$$ $$\left |z \right |=3 $$ $$phase= 90^{\circ}=\displaystyle\frac{\pi}{2}$$ ...
5
votes
1answer
117 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
2
votes
1answer
50 views

Factoring a complex polynomial

Factorize the polynomial : $$ p(x) = x^{5} - x^{4}+ 4x - 4 $$ In real factors in the lowest degree possible. So in previous questions I have been given at least one rot so that I can factorize it ...
1
vote
3answers
72 views

Complex Equations

The Equation: $$ z^{4} -2 z^{3} + 12z^{2} -14z + 35 = 0 $$ has a root with a real part 1, solve the equation. When it says a real part of 1, does this mean that we could use (z-1) and use ...
2
votes
2answers
89 views

Why is continuous differentiability required?

I have two questions. My book proves that if $f:\mathbb{C}\rightarrow \mathbb{C}$ is a holomorphic function, then it satisfies the Cauchy-Riemann equations, and if we look at the function as $F: ...
1
vote
3answers
89 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
5
votes
0answers
117 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
0
votes
1answer
34 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
3
votes
1answer
98 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
1
vote
1answer
29 views

complex functions inequalities plane

Given $w(z)=\frac{i-z}{i+z}$. Find the map w=f(z) of the part of the plane defined by inequalities: $|z|>1$ and $Im(z)>Re(z)$ so far: $|z|>1$ is this area from $Im(z)>Re(z)$ => ...
5
votes
1answer
46 views

how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$ If for all $y\in(y_0-\delta,y_0+\delta)$, ...
0
votes
1answer
71 views

Calculating Laurent Series of Complex Function

How does one alternate the Bernoulli number series expansion $$\frac x{e^x - 1}=\sum_{n=0}^{\infty}\frac{B_nx^n}{n!}$$ To calculate the Laurent Series centered at 0 in the annulus of convergence of ...
1
vote
1answer
31 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
0
votes
0answers
84 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
0
votes
1answer
84 views

Showing that derivative of conjugate is conjugate of derivative, using chain rule

I'm trying to show that the derivative of the conjugate is the conjugate of the derivative, i.e. $\dfrac{d[f(x)^*]}{dx} = [\dfrac{df(x)}{dx}]^*$, using the chain rule. Calling the conjugate * ...
2
votes
1answer
58 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
2
votes
1answer
45 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
1
vote
0answers
63 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
6
votes
2answers
178 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
0
votes
1answer
39 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
11
votes
0answers
230 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
2
votes
3answers
123 views

Integration by Euler's formula

How do you integrate the following by using Euler's formula, without using integration by parts? $$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}$$ I did integrate it by parts, by ...
0
votes
2answers
39 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?